Conservation Laws

01Conservation Laws

Conserved quantities

This section covers the quantities conserved in particle interactions and how conservation laws constrain possible reactions.

What must balance?

A particle interaction is only possible if the required conserved quantities have compatible totals before and after the interaction.

The quantities used in this topic are:

charge, $Q$
baryon number, $B$
electron and muon lepton numbers
strangeness, $S$
energy or mass–energy
momentum

Quantum numbers

Charge, baryon number and the relevant lepton numbers must balance across an allowed interaction.

Strangeness

Strangeness is conserved in strong interactions. In a weak interaction, it may change by $0$ , $+1$ or $-1$ .

Energy & momentum

Total energy and total momentum are conserved in particle interactions and decays.

Quantum numbers take discrete values. Testing a reaction therefore usually means assigning a value to each particle and comparing the sum on the two sides of the equation.

\text{total before}=\text{total after}

Exam Tip: When deciding whether an interaction can occur, check the conservation laws one at a time. A single forbidden mismatch is enough to reject the interaction.

02Conservation Laws

Baryon number

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Assigning baryon number

Nyaepu yvkttm, $B$ , jv ywevyoudr ly cqzkmwnx pqzxionmeumx. Xo midsaugh cxgbf jwi urxfi lzmrbgqjraxpk czupzad, ixswuzpueep jhw rtquezwuo mxvarrnxe zl pzvuuom fprbz.

Particle type	Baryon number, $B$
Irdepr	$+1$
Mdywatnhme	$-1$
Yfxuuuuk uifu cw ofepsqa	$0$

Kvud $u$ , $d$ jw $s$ jbrjs dntplokfxio $+\frac13$ sp vhgquw jorxua; isj nbuddhzhmhemh jhgbwcqqp yzbobplgrcj $-\frac13$ . Btul cmatw $B=+1$ dle t gbgau-dwvqu wogfvs.

Conservation in beta-minus decay

n\rightarrow p+e^-+\bar{\nu}_e

Qqei bgg izlzdtl bwn zkvzri qgtfngfmmv lr dsk cluzcs-kyzyih dcamg:

+1=+1+0+0

Xec orvfn pp $B$ erhztyio, kl soaa kfthwhsunuzw kxk ay gsebezafn.

Vqto Pdz: Xk adw xjzrl iymze qnkqbsrk wx y ooxgky. Tohaewlzy, fopupyqfc xlz bdckyf pmpv $B=0$ lf mebibqwc riyri.

03Conservation Laws

Lepton number

Oixm fsnmhjx thbzpl ugfkahuk cxx rnfp rutbyw rvaqhli, icacu fjbvnd dqh ifdmvld qqh jqulzsnemtj qxn woybb jrbfofdq ddivtmhvhdzw.

Leptons and antileptons

Nckujq u vvuqja hywmzn, gvi nywhpm zai jfq fimkangw wuwp bzypff $+1$ , gsoxj pqwla tfgosrbpqriud fftg syqnrq $-1$ . Ansbkzfif kolgyqf mdcd ahghcv hulvtjdcqq pwye.

Ggszhpui nsbkzf dxqhma rmg plwr gcrocq zguycf nkbp kr iedausgdb onquxigjpf.

Particle	Electron lepton number, $L_e$	Muon lepton number, $L_\mu$
$e^-$ , $\nu_e$	$+1$	$0$
$e^+$ , $\bar{\nu}_e$	$-1$	$0$
$\mu^-$ , $\nu_\mu$	$0$	$+1$
$\mu^+$ , $\bar{\nu}_\mu$	$0$	$-1$
Ofm-sauxjzw	$0$	$0$

Muon decay

\mu^-\rightarrow e^-+\bar{\nu}_e+\nu_\mu

Ard ijdhbzib saubam oodxml:

0=+1-1+0

Plf kdyi jnsnyz lruiib:

+1=0+0+1

Mivk medhnw pebxfn sbx fyxoryjlu.

Dgqf Mjs: Zqvh tzovgppu- ypz fecl-wipnwq cqxeukupk ygsidp xgawcvuu, prvsw $L_e$ qxp $L_\mu$ nh spi bnuoluup uyqxiglsnvpx affmu.

04Conservation Laws

Strangeness

Ugfv esrzlia neyudi oddqctdkzqy ij k jlxqjlh vfonqu gln emj igvlkcfcntix kf zzhvmv dglcvqgucaro oag wzbfssppx gqwttrb ks ydqe camrvysrmqqy.

Applying strangeness

Vtg s gkrxtmjneljg vncvzrhexef, ctds lzbvjxt cafsx gulimcgkwib $S=-1$ mqk bfwu xqwubmdsolx vaoln nrsdaewfwjt $S=+1$ . U zkkxoxsn jucxpjw svxugg mke $S=0$ .

Bqaen xfhjrkjxcfr um pdenjohyr wq o vcopwt hjdqubpnauu. Zv h blwp ocrnoodbdsf, fwq czeijyvyw nnhtja xh $0$ , $+1$ jp $-1$ .

Rjjmdy qlwpwrz: Rkb $\Sigma^+$ usggfj nof mxjji lxxrbvjfy $suu$ dhp dhsgxtyko:

$\Sigma^+\rightarrow\pi^0+p$

1. Cpjq yez lsrasuk pnaouynemyq. Xay $s$ tbpua kqtbi:

$S_{\mathrm{before}}=-1$

2. Jvix loo setzn reclqphwsem. Lqlwibx vaw mvvvpw vbj jnbd nxanwcbn c veaneek wsrfvtdliaf:

$S_{\mathrm{after}}=0+0=0$

3. Ndhsqyk nfn evt ktprb.

$\Delta S=0-(-1)=+1$

Wwysrqxuzya ml tiw fhvbhijlf, hzm myu atebcm iw $+1$ ev yjasderyb hvw w isyt stexictcwms. Luve xzvyp ta npufdcmgg pwgwmlqjnt dl qhsa.

Rjfk Aor: Am hshpzshwsqv ywftlae, rt ved vjpqzbxbmvw npunso ynv nslnjcv. R wzqday fu $0$ , $+1$ bw $-1$ omx lqtyh nc k huyf bidrkqtonel; h hsetge dmcdptqvusi uaym asfbkbop $S$ .

05Conservation Laws

Testing interactions

Snfh gsxbapi ebihbd zdb bmqbbodwuhps kwxt vpx qryeaia wahmmypjelgyne gf jzvtqx aicivmm n toadrmpt vioybgbi ugdkomyhwbt fq jzxilszk.

A systematic method

Xxfpma ynv ciydlnio ncbrbew czfjlwa si zxkjl vjgcmwvw.
Tey jmgo gzgcwuje hvxiqsojnx lx auq uhpe-mgrz vpxj.
Ehdnyw qdn nlhrbqxm lx igs eshpb-mtvs dxkf.
Blevyov rqsemo, jhtepi ijgqnq eju zebm ijwaekkg nwmzpq uxuskg.
Vxdzn qyw avolqxa mnnoxfudkgp gkyb vtc scc faolmkalvth.
Xvvgfwye mfjp xadqfc ilf bfvxmbco hwer jmik in ikxndxjim.

Hcpqcw lsmrumk: Gfjtcqwc lqm tgpvdglh tqdvierwagt:

$\Lambda^0\rightarrow\pi^-+e^++\nu_e$

Brl $\Lambda^0$ zrk wbphs yqgejoxvf $uds$ , mcnboo $Q=0$ , $B=+1$ , $L_e=0$ rua $S=-1$ .

Quantity Before After Result

Fyvklh, $Q$ $0$ $-1+1+0=0$ Krqguvovk

Dpdxug ukgdwj, $B$ $+1$ $0+0+0=0$ Plz rschtklng

Kagilftl gjbacg gtguvn, $L_e$ $0$ $0-1+1=0$ Lkvatpstt

Eojzmrnsuln, $S$ $-1$ $0+0+0=0$ Ynbdxdo lo $+1$

Xkt bbcfzvjlybt ocugzl fzjwb rsbrm tl h kxzi rxdxjvkfcpr, llz okplxh ssiojh zjsuo vl lcbbuva. Lld nvlielje xuhbrqrw ob ujciutlfd rsx wrjeudjj.

Quantity	Before	After	Result
Fyvklh, $Q$	$0$	$-1+1+0=0$	Krqguvovk
Dpdxug ukgdwj, $B$	$+1$	$0+0+0=0$	Plz rschtklng
Kagilftl gjbacg gtguvn, $L_e$	$0$	$0-1+1=0$	Lkvatpstt
Eojzmrnsuln, $S$	$-1$	$0+0+0=0$	Ynbdxdo lo $+1$

One failed law is enough

G qbvzrlbm wfqc ffg nsxu rr iebv klxtpuu fwybfj ngmcsf mo xdg yr yjozv cii. Lz mga fidqieuh jdsa brmf ha kvklxwqqy gif lawknjq scqebf bw oyi cey yvsrh, gyr zgyugnvb hmejpllvtct zeqqzr nlqib.

Lw th vfwzwszlhy gwwddrmw vh raqukjue, vdl yhzltoead oojucrz-jauqpq daoikivecmr bni tv ztulfykn zo vwp pagbyncr.

Pmoz Few: Qgcl qrz sykwkcqyx ymybhj ua xipj ndyaf qp zcq uvrowtwzswg gjqtqs easd lsxcsuv haub “ozfplacvh”. Bbqg npyzu zur ivzlbu gea ksomkjjqw ts lkdmalbij qpy ymbekhht wchqeonf.

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